Operators In Function Spaces

Research on operator theory in function space is always an important problem in the functional analysis. As a branch of mathematics, it has undergone a long history of study. Moreover, a complete and fruitful system of theory has been formed.Operator in different function space has different characteristics. Studies on operator briefly include boundedness, compactness, spectrum and algebraic( normal and subnormal ) property. Operator theory in classic spaces L~p and H~p, has formed its fruitful and complete theoretic system. As a spread of function space If, Orlicz space, formed in 1950's, is a particular class of function space with complex structure and fruitful content. It is combined with integral equation and successfully applied to other branches such as function approximation theory, partial differential equation and so on. Obviously, reseach on operator in Orlicz space becomes a meaningful work. However, Orlicz space is not similar to the classic function space L~p which has simple dual space. For this reason, the properties of some important operators, such as integral operator and composition operator in Orlicz space, are much more complex than those in L~p. Therefore, research on operators in this type of function space becomes more difficult.In this thesis, we focus on three types of operator-multiplication operator, composition operator and Toeplitz operator. Specifically, we discuss the following four parts:1. Multiplication operator in Orlicz space denned on finite measure set;2. Inclusions between Hardy-Orlicz spaces and multiplication operator in Hardy-Orlicz space;3. Composition operator in weighted Orlicz-Bergman space and weighted composition operator in Orlicz space;4. Toeplitz operator in Dirichlet space on multi-connected domain: compactness, spectrum and index formula of Toeplitz operator.Operators in several diffirent function spaces are discussed in this thesis. Firstly, in Orlicz space defined on finite measure set, we not only discuss some equivalent conditions about boundedness and metric-preserving property of multiplication operator, also discuss its compactness and spectrum; Secondly, we get some results about the inclusions between Hardy-Orlicz spaces, in which we find that equivalent conditions about compactness of multiplication operator is very similar to those in Orlicz space, and we also compute spectrum of multiplication operator; Thirdly, we mainly study composition operator ( weighted composition operator ) in weighted Orlicz-Bergman space ( or Orlicz space ). Similarly, we give some necessary and sufficient conditions about Fredholm property, invertibility, metric-preserving property of the dicussed operators in this thesis.Finally, we discuss Toeplitz operator in Dirichlet space on multi-connected domain, and compute the spectrum of Toeplitz operator in Dirichlet space on multi-connected domain, At the end of this paper, an index formula of Toeplitz operator is given.